6.04c Composite bodies: centre of mass

5 \includegraphics[max width=\textwidth, alt={}, center]{6b220343-1d64-4dbc-a42d-77967eef9c6d-08_449_890_262_630} \(O A B\) is a uniform lamina in the shape of a quadrant of a circle with centre \(O\) and radius 0.8 m which has its centre of mass at \(G\). The lamina is smoothly hinged at \(A\) to a fixed point and is free to rotate in a vertical plane. A horizontal force of magnitude 12 N acting in the plane of the lamina is applied to the lamina at \(B\). The lamina is in equilibrium with \(A G\) horizontal (see diagram).

1. Calculate the length \(A G\).
2. Find the weight of the lamina.

2 A uniform solid object is made by attaching a cone to a cylinder so that the circumferences of the base of the cone and a plane face of the cylinder coincide. The cone and the cylinder each have radius 0.3 m and height 0.4 m .

1. Calculate the distance of the centre of mass of the object from the vertex of the cone.
   [0pt] [The volume of a cone is \(\frac { 1 } { 3 } \pi r ^ { 2 } h\).]
   The object has weight \(W \mathrm {~N}\) and is placed with its plane circular face on a rough horizontal surface. A force of magnitude \(k W \mathrm {~N}\) acting at \(30 ^ { \circ }\) to the upward vertical is applied to the vertex of the cone. The object does not slip.
2. Find the greatest possible value of \(k\) for which the object does not topple.

5 A particle \(P\) of mass 0.7 kg is attached to a fixed point \(O\) by a light elastic string of natural length 0.6 m and modulus of elasticity 15 N . The particle \(P\) is projected vertically downwards from the point \(A , 0.8 \mathrm {~m}\) vertically below \(O\). The initial speed of \(P\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the distance below \(A\) of the point at which \(P\) comes to instantaneous rest.
2. Find the greatest speed of \(P\) in the motion. \includegraphics[max width=\textwidth, alt={}, center]{f922bf53-94a0-4ccc-8c38-959d2f795629-10_478_652_260_751} The diagram shows a uniform lamina \(A B C D E F G H\). The lamina consists of a quarter-circle \(O A B\) of radius \(r \mathrm {~m}\), a rectangle \(D E F G\) and two isosceles right-angled triangles \(C O D\) and \(G O H\). The rectangle has \(D G = E F = r \mathrm {~m}\) and \(D E = F G = x \mathrm {~m}\).
3. Given that the centre of mass of the lamina is at \(O\), express \(x\) in terms of \(r\).
4. Given instead that the rectangle \(D E F G\) is a square with edges of length \(r \mathrm {~m}\), state with a reason whether the centre of mass of the lamina lies within the square or the quarter-circle. \includegraphics[max width=\textwidth, alt={}, center]{f922bf53-94a0-4ccc-8c38-959d2f795629-12_384_693_258_726} A rough horizontal rod \(A B\) of length 0.45 m rotates with constant angular velocity \(6 \mathrm { rad } \mathrm { s } ^ { - 1 }\) about a vertical axis through \(A\). A small ring \(R\) of mass 0.2 kg can slide on the rod. A particle \(P\) of mass 0.1 kg is attached to the mid-point of a light inextensible string of length 0.6 m . One end of the string is attached to \(R\) and the other end of the string is attached to \(B\), with angle \(R P B = 60 ^ { \circ }\) (see diagram). \(R\) and \(P\) move in horizontal circles as the system rotates. \(R\) is in limiting equilibrium.
5. Show that the tension in the portion \(P R\) of the string is 1.66 N , correct to 3 significant figures.
6. Find the coefficient of friction between the ring and the rod.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

7 \includegraphics[max width=\textwidth, alt={}, center]{9daebcbe-826e-4eda-afa7-c935c6ea2bfc-10_451_574_258_781} \(A B C D\) is a uniform lamina in the shape of a trapezium which has centre of mass \(G\). The sides \(A D\) and \(B C\) are parallel and 1.8 m apart, with \(A D = 2.4 \mathrm {~m}\) and \(B C = 1.2 \mathrm {~m}\) (see diagram).

1. Show that the distance of \(G\) from \(A D\) is 0.8 m .
   The lamina is freely suspended at \(A\) and hangs in equilibrium with \(A D\) making an angle of \(30 ^ { \circ }\) with the vertical.
2. Calculate the distance \(A G\).
   With the lamina still freely suspended at \(A\) a horizontal force of magnitude 7 N acting in the plane of the lamina is applied at \(D\). The lamina is in equilibrium with \(A G\) making an angle of \(10 ^ { \circ }\) with the downward vertical.
3. Find the two possible values for the weight of the lamina.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

6 \includegraphics[max width=\textwidth, alt={}, center]{add3948c-3b45-4e67-ac84-e2ca935afd64-08_442_953_237_596} An object is formed by joining a hemispherical shell of radius 0.2 m and a solid cone with base radius 0.2 m and height \(h \mathrm {~m}\) along their circumferences. The centre of mass, \(G\), of the object is \(d \mathrm {~m}\) from the vertex of the cone on the axis of symmetry of the object. The object rests in equilibrium on a horizontal plane, with the curved surface of the cone in contact with the plane (see diagram). The object is on the point of toppling.

1. Show that \(d = h + \frac { 0.04 } { h }\).
2. It is given that the cone is uniform and of weight 4 N , and that the hemispherical shell is uniform and of weight \(W \mathrm {~N}\). Given also that \(h = 0.8\), find \(W\).

3 One end of a light elastic string, of natural length \(a\) and modulus of elasticity \(4 m g\), is attached to a fixed point \(O\). The other end of the string is attached to a particle of mass \(m\). The particle moves in a horizontal circle with a constant angular speed \(\sqrt { \frac { \mathrm { g } } { \mathrm { a } } }\) with the string inclined at an angle \(\theta\) to the downward vertical through \(O\). The length of the string during this motion is \(( \mathrm { k } + 1 ) \mathrm { a }\).

1. Find the value of \(k\).
2. Find the value of \(\cos \theta\). \includegraphics[max width=\textwidth, alt={}, center]{1c53c407-25ea-43fc-a571-74ba1fffea8f-06_584_695_264_667} The diagram shows the cross-section \(A B C D\) of a uniform solid object which is formed by removing a cone with cross-section \(D C E\) from the top of a larger cone with cross-section \(A B E\). The perpendicular distance between \(A B\) and \(D C\) is \(h\), the diameter \(A B\) is \(6 r\) and the diameter \(D C\) is \(2 r\).
   1. Find an expression, in terms of \(h\), for the distance of the centre of mass of the solid object from \(A B\).
      The object is freely suspended from the point \(B\) and hangs in equilibrium. The angle between \(A B\) and the downward vertical through \(B\) is \(\theta\).
   2. Given that \(h = \frac { 13 } { 4 } r\), find the value of \(\tan \theta\).

3 A light elastic string has natural length \(a\) and modulus of elasticity 12 mg . One end of the string is attached to a fixed point \(O\). The other end of the string is attached to a particle of mass \(m\). The particle hangs in equilibrium vertically below \(O\). The particle is pulled vertically down and released from rest with the extension of the string equal to \(e\), where \(\mathrm { e } > \frac { 1 } { 3 } \mathrm { a }\). In the subsequent motion the particle has speed \(\sqrt { 2 \mathrm { ga } }\) when it has ascended a distance \(\frac { 1 } { 3 } a\). Find \(e\) in terms of \(a\). \includegraphics[max width=\textwidth, alt={}, center]{b10c65ef-dafd-4746-be5b-789130b7d030-06_488_496_269_781} A uniform lamina \(A E C F\) is formed by removing two identical triangles \(B C E\) and \(C D F\) from a square lamina \(A B C D\). The square has side \(3 a\) and \(E B = D F = h\) (see diagram).

1. Find the distance of the centre of mass of the lamina \(A E C F\) from \(A D\) and from \(A B\), giving your answers in terms of \(a\) and \(h\).
   The lamina \(A E C F\) is placed vertically on its edge \(A E\) on a horizontal plane.
2. Find, in terms of \(a\), the set of values of \(h\) for which the lamina remains in equilibrium.

2 A light elastic string has natural length \(a\) and modulus of elasticity 4 mg . One end of the string is fixed to a point \(O\) on a smooth horizontal surface. A particle \(P\) of mass \(m\) is attached to the other end of the string. The particle \(P\) is projected along the surface in the direction \(O P\). When the length of the string is \(\frac { 5 } { 4 } a\), the speed of \(P\) is \(v\). When the length of the string is \(\frac { 3 } { 2 } a\), the speed of \(P\) is \(\frac { 1 } { 2 } v\).

1. Find an expression for \(v\) in terms of \(a\) and \(g\).
2. Find, in terms of \(g\), the acceleration of \(P\) when the stretched length of the string is \(\frac { 3 } { 2 } a\). \includegraphics[max width=\textwidth, alt={}, center]{5e95e0c9-d47d-4f2b-89da-ab949b9661f4-04_552_1059_264_502} A smooth cylinder is fixed to a rough horizontal surface with its axis of symmetry horizontal. A uniform rod \(A B\), of length \(4 a\) and weight \(W\), rests against the surface of the cylinder. The end \(A\) of the rod is in contact with the horizontal surface. The vertical plane containing the rod \(A B\) is perpendicular to the axis of the cylinder. The point of contact between the rod and the cylinder is \(C\), where \(A C = 3 a\). The angle between the rod and the horizontal surface is \(\theta\) where \(\tan \theta = \frac { 3 } { 4 }\) (see diagram). The coefficient of friction between the rod and the horizontal surface is \(\frac { 6 } { 7 }\). A particle of weight \(k W\) is attached to the rod at \(B\). The rod is about to slip. The normal reaction between the rod and the cylinder is \(N\).

2 A ball of mass 2 kg is projected vertically downwards with speed \(5 \mathrm {~ms} ^ { - 1 }\) through a liquid. At time \(t \mathrm {~s}\) after projection, the velocity of the ball is \(v \mathrm {~ms} ^ { - 1 }\) and its displacement from its starting point is \(x \mathrm {~m}\). The forces acting on the ball are its weight and a resistive force of magnitude \(0.2 v ^ { 2 } \mathrm {~N}\).

1. Find an expression for \(v\) in terms of \(t\).
2. Deduce what happens to \(v\) for large values of \(t\). \includegraphics[max width=\textwidth, alt={}, center]{e7091f6c-af72-49f3-b825-cdce9fb2c06f-06_803_652_251_703} A uniform square lamina of side \(2 a\) and weight \(W\) is suspended from a light inextensible string attached to the midpoint \(E\) of the side \(A B\). The other end of the string is attached to a fixed point \(P\) on a rough vertical wall. The vertex \(B\) of the lamina is in contact with the wall. The string \(E P\) is perpendicular to the side \(A B\) and makes an angle \(\theta\) with the wall (see diagram). The string and the lamina are in a vertical plane perpendicular to the wall. The coefficient of friction between the wall and the lamina is \(\frac { 1 } { 2 }\). Given that the vertex \(B\) is about to slip up the wall, find the value of \(\tan \theta\). \includegraphics[max width=\textwidth, alt={}, center]{e7091f6c-af72-49f3-b825-cdce9fb2c06f-08_581_576_269_731} A light elastic string has natural length \(8 a\) and modulus of elasticity \(5 m g\). A particle \(P\) of mass \(m\) is attached to the midpoint of the string. The ends of the string are attached to points \(A\) and \(B\) which are a distance \(12 a\) apart on a smooth horizontal table. The particle \(P\) is held on the table so that \(A P = B P = L\) (see diagram). The particle \(P\) is released from rest. When \(P\) is at the midpoint of \(A B\) it has speed \(\sqrt { 80 a g }\).
   1. Find \(L\) in terms of \(a\).
   2. Find the initial acceleration of \(P\) in terms of \(g\).

4. \begin{figure}[h] \end{figure} The uniform plane lamina \(A B C D E F\) shown in Figure 1 is made from two identical rhombuses. Each rhombus has sides of length \(a\) and angle \(B A D =\) angle \(D A F = \theta\). The centre of mass of the lamina is \(0.9 a\) from \(A\).

1. Show that \(\cos \theta = 0.8\) The weight of the lamina is \(W\). A particle of weight \(k W\) is fixed to the lamina at the point \(A\). The lamina is freely suspended from \(B\) and hangs in equilibrium with \(D A\) horizontal.
2. Find the value of \(k\).

2. \begin{figure}[h] \end{figure} A uniform lamina is in the shape of a trapezium \(A B C D\) with \(A B = a , D A = D C = 2 a\) and angle \(B A D =\) angle \(A D C = 90 ^ { \circ }\), as shown in Figure 1. The centre of mass of the lamina is at the point \(G\).

1. 1. Show that the distance of \(G\) from \(A B\) is \(\frac { 10 a } { 9 }\).
   2. Find the distance of \(G\) from \(A D\). The mass of the lamina is \(3 M\). A particle of mass \(k M\) is now attached to the lamina at \(B\). The lamina is freely suspended from the midpoint of \(A D\) and hangs in equilibrium with \(A D\) horizontal.
2. Find the value of \(k\).

3. \begin{figure}[h] \end{figure} The uniform lamina \(O A B C D\) is shown in Figure 1, with \(O A = 6 a , A B = 3 a , C D = 2 a\) and \(D O = 6 a\) and with right angles at \(O , A\) and \(D\).

1. Find the distance of the centre of mass of the lamina
   1. from \(O D\),
   2. from \(O A\). The lamina is suspended from \(C\) and hangs freely in equilibrium with \(C B\) inclined at an angle \(\alpha\) to the vertical.
2. Find, to the nearest degree, the size of the angle \(\alpha\).

4. \begin{figure}[h] \end{figure} The uniform lamina \(L\), shown shaded in Figure 1, is formed by removing a circular disc of radius \(2 a\) from a uniform circular disc of radius \(4 a\). The larger disc has centre \(O\) and diameter \(A B\). The radius \(O D\) is perpendicular to \(A B\). The smaller disc has centre \(C\), where \(C\) is on \(A B\) and \(B C = 3 a\)

1. Show that the centre of mass of \(L\) is \(\frac { 13 } { 3 } a\) from \(B\). The mass of \(L\) is \(M\) and a particle of mass \(k M\) is attached to \(L\) at \(B\). When \(L\), with the particle attached, is freely suspended from point \(D\), it hangs in equilibrium with \(A\) higher than \(B\) and \(A B\) at an angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 3 } { 4 }\)
2. Find the value of \(k\).

4. [The centre of mass of a uniform semicircular lamina of radius \(r\) is \(\frac { 4 r } { 3 \pi }\) from the centre.] \begin{figure}[h] \end{figure} The uniform rectangular lamina \(A B C D E F\) has sides \(A C = F D = 6 a\) and \(A F = C D = 3 a\). The point \(B\) lies on \(A C\) with \(A B = 2 a\) and the point \(E\) lies on \(F D\) with \(F E = 2 a\). The template, \(T\), shown shaded in Figure 3, is formed by removing the semicircular lamina with diameter \(B C\) from the rectangular lamina and then fixing this semicircular lamina to the opposite side, \(F D\), of the rectangular lamina. The diameter of the semicircular lamina coincides with \(E D\) and the semicircular arc \(E D\) is outside the rectangle \(A B C D E F\). All points of \(T\) lie in the same plane.

1. Show that the centre of mass of \(T\) is a distance \(\left( \frac { 9 + 2 \pi } { 6 } \right)\) a from \(A C\). The mass of \(T\) is \(M\). A particle of mass \(k M\) is attached to \(T\) at \(C\). The loaded template is freely suspended from \(A\) and hangs in equilibrium with \(A F\) at angle \(\phi\) to the downward vertical through \(A\). Given that \(\tan \phi = \frac { 3 } { 2 }\)
2. find the value of \(k\).
   \section*{\textbackslash section*\{Question 4 continued\}} \includegraphics[max width=\textwidth, alt={}, center]{c16c17b6-2c24-4939-b3b5-63cd63646b76-11_149_142_2604_1816}

2. \begin{figure}[h] \end{figure} The uniform lamina \(A B C\) has sides \(A B = A C = 13 a\) and \(B C = 10 a\). The lamina is freely suspended from \(A\). A horizontal force of magnitude \(F\) is applied to the lamina at \(B\), as shown in Figure 1. The line of action of the force lies in the vertical plane containing the lamina. The lamina is in equilibrium with \(A B\) vertical. The weight of the lamina is \(W\). Find \(F\) in terms of \(W\).
VILM SIHI NI JAIUM ION OC
VANV SIHI NI I III M LON OO
VI4V SIHI NI JAIUM ION OC

4. \begin{figure}[h] \end{figure} The number "4", shown in Figure 2, is a rigid framework made from three uniform rods, \(A B , B C\) and \(C D\), where $$A B = 6 a , B C = 5 a \text { and } C D = 4 a$$ The point \(E\) is on \(A B\) and \(C D\), where \(B E = 4 a , C E = 3 a\) and angle \(C E B = 90 ^ { \circ }\) The three rods are all made from the same material and they all lie in the same plane. The framework is suspended from \(B\) and hangs in equilibrium with \(B A\) at an angle \(\theta\) to the downward vertical. Find \(\theta\) to the nearest degree.
VILU SIHI NI JAIUM ION OC
VIUV SIHI NI JAHM ION OC
VIIV SIHI NI EIIIM ION OC

1. \hspace{0pt} [The centre of mass of a semicircular arc of radius \(r\) is \(\frac { 2 r } { \pi }\) from the centre.]

\begin{figure}[h] \end{figure} Uniform wire is used to form the framework shown in Figure 2.
In the framework,

• \(A B C\) is straight and has length \(25 a\)
• \(A D E\) is straight and has length \(24 a\)
• \(A B D\) is a semicircular arc of radius \(7 a\)
• \(E C = 7 a\)
• angle \(A E C = 90 ^ { \circ }\)
• the points \(A , B , C , D\) and \(E\) all lie in the same plane

The distance of the centre of mass of the framework from \(A E\) is \(d\).

1. Show that \(d = \frac { 53 } { 2 ( 7 + \pi ) } a\) The framework is freely suspended from \(A\) and hangs in equilibrium with \(A C\) at angle \(\alpha ^ { \circ }\) to the downward vertical.
2. Find the value of \(\alpha\).

3. \begin{figure}[h] \end{figure} The uniform lamina \(A B D E\) is in the shape of a rectangle with \(A B = 8 a\) and \(B D = 6 a\). The triangle \(B C D\) is isosceles and has base \(6 a\) and perpendicular height \(6 a\). The template \(A B C D E\), shown shaded in Figure 1, is formed by removing the triangular lamina \(B C D\) from the lamina \(A B D E\).

1. Show that the centre of mass of the template is \(\frac { 14 } { 5 } a\) from \(A E\). The template is freely suspended from \(A\) and hangs in equilibrium with \(A B\) at an angle of \(\theta ^ { \circ }\) to the downward vertical.
2. Find the value of \(\theta\), giving your answer to the nearest whole number.

5. \begin{figure}[h] \end{figure} A beam \(A B\) has mass 12 kg and length 5 m . It is held in equilibrium in a horizontal position by two vertical ropes attached to the beam. One rope is attached to \(A\), the other to the point \(C\) on the beam, where \(B C = 1 \mathrm {~m}\), as shown in Figure 2. The beam is modelled as a uniform rod, and the ropes as light strings.

1. Find
   1. the tension in the rope at \(C\),
   2. the tension in the rope at \(A\). A small load of mass 16 kg is attached to the beam at a point which is \(y\) metres from \(A\). The load is modelled as a particle. Given that the beam remains in equilibrium in a horizontal position,
2. find, in terms of \(y\), an expression for the tension in the rope at \(C\). The rope at \(C\) will break if its tension exceeds 98 N. The rope at \(A\) cannot break.
3. Find the range of possible positions on the beam where the load can be attached without the rope at \(C\) breaking.

4. \begin{figure}[h] \end{figure} A bench consists of a plank which is resting in a horizontal position on two thin vertical legs. The plank is modelled as a uniform rod \(P S\) of length 2.4 m and mass 20 kg . The legs at \(Q\) and \(R\) are 0.4 m from each end of the plank, as shown in Figure 1. Two pupils, Arthur and Beatrice, sit on the plank. Arthur has mass 60 kg and sits at the middle of the plank and Beatrice has mass 40 kg and sits at the end \(P\). The plank remains horizontal and in equilibrium. By modelling the pupils as particles, find

1. the magnitude of the normal reaction between the plank and the leg at \(Q\) and the magnitude of the normal reaction between the plank and the leg at \(R\). Beatrice stays sitting at \(P\) but Arthur now moves and sits on the plank at the point \(X\). Given that the plank remains horizontal and in equilibrium, and that the magnitude of the normal reaction between the plank and the leg at \(Q\) is now twice the magnitude of the normal reaction between the plank and the leg at \(R\),
2. find the distance \(Q X\).

4. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} The uniform rectangular lamina \(A B C D\), shown in Figure 2, has \(D C = 4 a\) and \(A D = 5 a\) The points \(S\) on \(A B\) and \(T\) on \(B C\) are such that \(S B = B T = 3 a\) The lamina is folded along \(S T\) to form the folded lamina \(L\), shown in Figure 3.
The distance of the centre of mass of \(L\) from \(A D\) is \(d\).

1. Show that \(d = \frac { 71 } { 40 } a\) The weight of \(L\) is \(4 W\). A particle of weight \(W\) is attached to \(L\) at \(C\).
   The folded lamina \(L\) is freely suspended from \(S\).
   A force of magnitude \(F\), acting parallel to \(D C\), is applied to \(L\) at \(D\) so that \(A D\) is vertical.
2. Find \(F\) in terms of \(W\)

4. \begin{figure}[h] \end{figure} The uniform square lamina \(A B C D\) shown in Figure 2 has sides of length 4a. The points \(E\) and \(F\), on \(D A\) and \(D C\) respectively, are both at a distance \(3 a\) from \(D\). The portion \(D E F\) of the lamina is folded through \(180 ^ { \circ }\) about \(E F\) to form the folded lamina \(A B C F E\) shown in Figure 3 below. \begin{figure}[h] \end{figure}

1. Show that the distance from \(A B\) of the centre of mass of the folded lamina is \(\frac { 55 } { 32 } a\).
   (6) The folded lamina is freely suspended from \(E\) and hangs in equilibrium.
2. Find the size of the angle between \(E D\) and the downward vertical.

4. \begin{figure}[h] \end{figure} The uniform lamina \(A B C D E\) is made by joining a rectangular lamina \(A B D E\) to a triangular lamina \(B C D\) along the edge \(B D\). The rectangle has length \(6 a\) and width \(3 a\). The triangle is isosceles, with \(B C = C D\), and the distance from \(C\) to \(B D\) is \(3 a\), as shown in Figure 2.

1. Find the distance of the centre of mass of the lamina, \(A B C D E\), from \(A E\). The lamina \(A B C D E\) is freely suspended from \(A\). A horizontal force of magnitude \(F\) newtons is applied to the lamina at \(D\). The line of action of the force lies in the vertical plane containing the lamina. The lamina is in equilibrium with \(A E\) vertical. The mass of the lamina is 4 kg .
2. Find the magnitude of the force exerted on the lamina at \(A\).
   DO NOT WIRITE IN THIS AREA

6. \begin{figure}[h] \end{figure} The uniform lamina \(A B C D\) is a square with sides of length \(6 a\). The point \(E\) is the midpoint of side \(A D\). The triangle \(C D E\) is removed from the square to form the uniform lamina \(L\), shown in Figure 3. The centre of mass of \(L\) is at the point \(G\).

1. Show that the distance of \(G\) from the side \(A B\) is \(\frac { 7 } { 3 } a\).
2. Find the distance of \(G\) from the side \(A E\). The mass of \(L\) is \(M\). A particle of mass \(k M\) is attached to \(L\) at the point \(E\). The lamina, with the particle attached, is freely suspended from \(A\) and hangs in equilibrium with the diagonal \(A C\) vertical.
3. Find the value of \(k\).

4. \begin{figure}[h] \end{figure} The uniform lamina \(A B C\) is an isosceles triangle with \(A B = B C , A C = 6 a\) and the distance from \(B\) to \(A C\) is \(3 a\). The uniform lamina \(M N C\) is an isosceles triangle with \(M N = N C\) and \(M C = 3 a\). Triangles \(A B C\) and \(M N C\) are similar and are made of the same material. The lamina \(L\) is formed by fixing triangle \(M N C\) on top of triangle \(A B C\), as shown in Figure 2.

1. Show that the distance of the centre of mass of \(L\) from \(A C\) is \(\frac { 9 } { 10 } a\) The lamina \(L\) is freely suspended from \(B\) and hangs in equilibrium.
2. Find, to the nearest degree, the size of the angle between \(A B\) and the downward vertical.